A numeraire is a traded asset that can be used as relative measure of value.In physics, quantities can be measured in units which are constant and well defined, meters or seconds for example. Unfortunately in finance the measurement of value is a little trickier. For example a cheque for $100,000 is worth maybe £61,000 today but could be worth £60,000 pounds next week because of exchange rate fluctuations. It might buy me a car now, but it won't buy me a car in fifty years time because of inflation. The problem is that the dollar is itself a traded asset. The solution is to model the behaviour of a traded asset and to use this as a relative unit of value.As an example consider the traditional numeraire, the zero coupon bond which guarantees to pay (say) $100 in one-years time.I know that today the bond can be bought for $50. My cheque is worth 2000*B, or two-thousand zero coupon bonds. In one years time it will only be worth 1000*B, its relative value is less even though its apparent value hasn't changed (money now is better than money later). Here expressing everything in units of zero-coupon bonds allows values assets at different times to be compared.Of course I can use any traded asset as a relative measure of value, but this is just a change in the way I make measurements so it shouldn't affect the real value of things. This is the principle of numeraire invariance and it is a very fundamental symmetry of finance which should be observed by any mathematical models. Being able to change units in a calculation is sometimes a very effective way of simplifying the calculation, however it is very important to remember that the pricing measure is the measure which makes all the assets written in units of the numeraire assets martingales. This means that the measure will be different if we decide to use a different numeraire. One simple example is the pricing of an option to exchange one share for another share at a future date. Measured in pounds the payoff of this option is (£)max(S_1(T)-S_2(T),0) where the assets satisfy the equations (usual Black-Scholes terminology),S_i(t)=S_i(0)exp[(mu_i-1/2 sigma_i^2)t+sigma_i w_i] (Log-Normal) and [dw_1, dw_2]=rho dtCalculating the option price requires the calculation of the expected value over both correllated brownian motions, which is a bit tricky. However, measured in units of the second stock price (at the appropriate time) the payoff is max(S_1(T)/S_2(T)-1,0)Now Y=S_1/S_2 is easy because of the Log-Normal distributions and shows that Y is also Log-Normal with a volatility sqrt(sigma_1^2+sigma_2^2-2rho sigma_1 sigma_2). The option price formula is (in units of S_2) E[max(Y-1,0)] or in pounds (S_2)(0)*E[max(Y-1,0)], where the expected value is taken under the measure which makes Y a martingale, which is just S_2*BS(S=S_1/S_2, K=1, r=0,sigma=sqrt(sigma_1^2+sigma_2^2-2rho sigma_1 sigma_2), T=T).This is known as the Margrabe formula.Im sure lots of people will provide lots of other examples in the rest of the thread.
Last edited by spacemonkey
on October 4th, 2003, 10:00 pm, edited 1 time in total.